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A051523 Generalized Stirling number triangle of first kind. 6
1, -10, 1, 110, -21, 1, -1320, 362, -33, 1, 17160, -6026, 791, -46, 1, -240240, 101524, -17100, 1435, -60, 1, 3603600, -1763100, 358024, -38625, 2335, -75, 1, -57657600, 31813200, -7491484, 976024, -75985, 3535, -91, 1, 980179200, -598482000, 159168428, -24083892, 2267769, -136080, 5082, -108, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n,m)= ^10P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(10+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(10*t),exp(t)-1).

REFERENCES

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

FORMULA

a(n, m)= a(n-1, m-1) - (n+9)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n<m; a(n, -1) := 0, a(0, 0)=1.

E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^10).

Triangle (signed) = [ -10, -1, -11, -2, -12, -3, -13, -14, -4, ...] DELTA A000035; triangle (unsigned) = [10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,10), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

EXAMPLE

{1}; {-10,1}; {110,-21,1}; {-1320,362,-331}; ... s(2,x)= 110-21*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

PROG

(Haskell)

a051523 n k = a051523_tabl !! n !! k

a051523_row n = a051523_tabl !! n

a051523_tabl = map fst $ iterate (\(row, i) ->

   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 10)

-- Reinhard Zumkeller, Mar 12 2014

CROSSREFS

The first (m=0) unsigned column sequence is A049398. Row sums (signed triangle): A049389(n)*(-1)^n. Row sums (unsigned triangle): A051431(n).

See also A049444 A049458 A049459 A049460 A051338 A051339 A051379 A051380

Cf. A000035 A084938.

Sequence in context: A130310 A288050 A288503 * A181868 A287494 A287753

Adjacent sequences:  A051520 A051521 A051522 * A051524 A051525 A051526

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified January 22 23:00 EST 2019. Contains 319365 sequences. (Running on oeis4.)